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(Covid-19) in Predicting Future Behaviours and Sensitivity Analysis

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(Covid-19) in Predicting Future Behaviours and Sensitivity Analysis Math. Model. Nat. Phenom. 15 (2020) 33 Mathematical Modelling of Natural Phenomena https://doi.org/10.1051/mmnp/2020020 www.mmnp-journal.org MATHEMATICAL MODELLING FOR CORONAVIRUS DISEASE (COVID-19) IN PREDICTING FUTURE BEHAVIOURS AND SENSITIVITY ANALYSIS Sarbaz H. A. Khoshnaw1,*, Rizgar H. Salih1 and Sadegh Sulaimany2 Abstract. Nowadays, there are a variety of descriptive studies of available clinical data for coron- avirus disease (COVID-19). Mathematical modelling and computational simulations are effective tools that help global efforts to estimate key transmission parameters. The model equations often require computational tools and dynamical analysis that play an important role in controlling the disease. This work reviews some models for coronavirus first, that can address important questions about the global health care and suggest important notes. Then, we model the disease as a system of differential equations. We develop previous models for the coronavirus, some key computational simulations and sensitivity analysis are added. Accordingly, the local sensitivities for each model state with respect to the model parameters are computed using three different techniques: non-normalizations, half nor- malizations and full normalizations. Results based on sensitivity analysis show that almost all model parameters may have role on spreading this virus among susceptible, exposed and quarantined sus- ceptible people. More specifically, communicate rate person{to{person, quarantined exposed rate and transition rate of exposed individuals have an effective role in spreading this disease. One possible solution suggests that healthcare programs should pay more attention to intervention strategies, and people need to self-quarantine that can effectively reduce the disease. Mathematics Subject Classification. 92D30,92D25,92C42, 34C60. Received March 30, 2020. Accepted May 12, 2020. 1. Introduction Understanding and predicting novel coronavirus (COVID-19) has become very important owing to the huge global health burden. Till 20 March 2020, a surge of coronavirus disease cases reached over 168 countries and territories around the world with more than 210 000 cases including near to 9000 deaths [18]. The increasing worldwide infection of coronavirus disease, it is very important to know it's dynamic as soon and as much as possible. Recently, several mathematical, computational, clinical and examination studies have been put forward for modeling, prediction, treatment and control of the disease, there is still room for improvement. In recent days, modeling novel coronavirus disease has become of extreme importance and several mathematical Keywords and phrases: Coronavirus disease (COVID-19), mathematical modeling, sensitivity analysis, differential equations, computational simulations. 1 Departemnt of Mathematics, College of Basic Education, University of Raparin, Ranya, Kurdistant Region of Iraq. 2 Department of Computer Engineering, University of Kurdistan, Sanandaj, Iran. * Corresponding author: [email protected] c The authors. Published by EDP Sciences, 2020 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 S.H.A. KHOSHNAW ET AL. and computational researches have been proposed for the prediction of the disease dynamics. Here, we avoid reviewing related preprint papers because of their huge number and lack of formal review evaluation and acceptance by valid journals. One approach proposed by Muhammad and Abdon [5] involved the use of fractional derivative for infection minimization. They considered the seafood market as the main source of infection when the bats and the unknown hosts leave the infection there. After reducing the model into the seafood market, and formulating a fractional model, they parameterized the model using January 2020 data cases. The basic reproduction data calculated by the study was 2.4829. Sha et al. developed a discrete{time stochastic epidemic model with binomial distributions to study the transmission of the disease [4]. Model parameters are estimated on the basis of fitting to recently reported data from the National Health Commission of the People's Republic of China. Their simulation showed that confirmed cases of the disease will reach the peak near to the end of February 2020 for the country. Tang et al. in [15] constructed a deterministic compartmental model based on the clinical progression of the disease, epidemiological status of the individuals, and intervention measures. Their estimations show that the control reproduction number may be as high as 6.47. It can be reduced by interventions such as isolation and quarantine, according to a sensitivity analysis. In another study given in [10] that combined a stochastic transmission model with COVID-19 case data from and originated from Wuhan. They modelled transmission as a geometric random walk process, and they used sequential Monte Carlo simulation to infer the transmission rate over time. Based on their calculations, disease transmission probably stopped in Wuhan during late January 2020, but the cases that arrive at international locations from Wuhan, may lead to epidemics eventually. Chen et al. developed a simplified model as Reservoir{People (RP) transmission network and adopted next generation matrix approach to calculate the basic reproduction number to assess the transmissibility of the SARS-CoV-2 [3]. According to their finding, the expected number of secondary infections was 3.58. Tang et al. updated their previous estimation of the transmission risk of the novel coronavirus, based on the recent advances of technological improvement for detection and confirmation of new infected cases [16]. Their calculations were based on a time{dependent dynamic model of contact and diagnose rate, that led to re{estimated daily reproduction number; they observed that the effective daily reproduction ratio has already fallen below 1, end of January 2020. Another mathematical model for this new virus is the efficiency of quarantine measures. The assumption is that all infected individual are isolated after the incubation period. The basic reproduction number R0 has an effective role in disease progression [17]. Although numerous modelling methods have been projected so far for new coronavirus disease prediction, a lot can still be improved. Defining models as mass action law with reaction rate constants and calculating the sensitivities for each model state with respect to model parameters, could improve the outcomes. An issue that has not been investigated enough is various local sensitivity analysis for non-normalizations, half normalizations and full normalizations techniques in computational simulations for COVID-19. In a complicated modeling case like new coronavirus dynamics, it is necessary to pay attention to sensitivity analysis more accurately and widely. Even though few related papers used sensitivity analysis in their researches [12, 15], the usage was only limited to few special variables and cases. Another novelty of the paper is a simplification of the identification of the critical model parameters, which makes it easy to be used by biologists with less knowledge of mathematical modeling and also facilitates the improvement of the model for future development. 2. Mathematical modeling for coronavirus disease There are many models for describing the spread of infectious diseases. The well defined model in this field is \Susceptible{Exposed{Infectious{Recovered" model. This is sometimes called SEIR model [11]. The main idea of the model is based on the clinical progression, epidemiological individuals and intervention measures. Accordingly, the infectious disease SEIR model has been investigated, the model combined with the intervention compartments such as treatment, isolation (hospitalization) and quarantine. We develop the previous model of coronavirus disease (COVID-19) given in [15, 16], the model diagram and the interaction individual components given in Figure1. The model initial populations and interaction parameters are obtained for the confirmed cases in China. MATHEMATICAL MODELLING FOR CORONAVIRUS DISEASE (COVID-19) 3 Figure 1. The model interaction individuals for the coronavirus disease (COVID-19) with reaction rates. The model reactions with their rates are shown below: v1 v2 v3 v4 S−!E; S−!Eq;S−!Sq:Sq−!S; E−!v5 A; E−!v6 I;E −!v7 H; A−!v8 R; q (2.1) I−!v9 R; I−!v10 H; I−!v11 null;H−!v12 R; H−!v13 null: The model populations (variables) with their biological descriptions are given in Table1. Furthermore, the model constant interactions (parameters) with their definitions are described in Table2. Table 1. The model variables for coronavirus disease (COVID-19) with their biological meaning. No. Variables Biological Descriptions 1 S Susceptible Individuals 2 E Exposed Individuals 3 I Symptomatic Infected Individuals 4 A Pre-symptomatic Individuals 5 Sq Quarantined Susceptible Individuals 6 Eq Quarantined Exposed Individuals 7 H Quarantined Infected (Hospitalized) Individuals 8 R Recovered Individuals 4 S.H.A. KHOSHNAW ET AL. Table 2. The model reaction constants (parameters) and initial individual populations for COVID-19 epidemic outbreak with their biological definitions, all data are confirmed cases in China presented in [15]. Symbols Biological definitions Estimated values S(0) Initial susceptible individuals 11:081
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